In mathematics, Hadamard regularization (also called Hadamard finite part or Hadamard's partie finie) is a method of regularizing divergent integrals by dropping some divergent terms and keeping the finite part, introduced by Jacques Hadamard (1923, book III, chapter I, 1932). Marcel Riesz (1938, 1949) showed that this can be interpreted as taking the meromorphic continuation of a convergent integral.
Description If the Cauchy principal value integral
C ∫ a b f ( t ) t − x d t ( for a < x < b ) {\displaystyle {\mathcal {C}}\int _{a}^{b}{\frac {f(t)}{t-x}}\,dt\quad ({\text{for }}a<x<b)}
exists, then it may be differentiated with respect to x to obtain the Hadamard finite part integral as follows:
d d x ( C ∫ a b f ( t ) t − x d t ) = H ∫ a b f ( t ) ( t − x ) 2 d t ( for a < x < b ) . {\displaystyle {\frac {d}{dx}}\left({\mathcal {C}}\int _{a}^{b}{\frac {f(t)}{t-x}}\,dt\right)={\mathcal {H}}\int _{a}^{b}{\frac {f(t)}{(t-x)^{2}}}\,dt\quad ({\text{for }}a<x<b).}
Note that the symbols C {\displaystyle {\mathcal {C}}} and H {\displaystyle {\mathcal {H}}} are used here to denote Cauchy principal value and Hadamard finite-part integrals respectively. The Hadamard finite part integral above (for a < x < b) may also be given by the following equivalent definitions:
H ∫ a b f ( t ) ( t − x ) 2 d t = lim ε → 0 + { ∫ a x − ε f ( t ) ( t − x ) 2 d t + ∫ x + ε b f ( t ) ( t − x ) 2 d t − f ( x + ε ) + f ( x − ε ) ε } , {\displaystyle {\mathcal {H}}\int _{a}^{b}{\frac {f(t)}{(t-x)^{2}}}\,dt=\lim _{\varepsilon \to 0^{+}}\left\{\int _{a}^{x-\varepsilon }{\frac {f(t)}{(t-x)^{2}}}\,dt+\int _{x+\varepsilon }^{b}{\frac {f(t)}{(t-x)^{2}}}\,dt-{\frac {f(x+\varepsilon )+f(x-\varepsilon )}{\varepsilon }}\right\},}
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